Find the length of the longest rod that can be placed in a small box with length 20 cm , breadth 20 cm and height =10 cm

Answer

To find the length of the longest rod that can be placed in a box with given dimensions, we will use the formula for the diagonal of a cuboid. The diagonal \(d\) can be calculated using the formula:

\[ d = \sqrt{L^2 + B^2 + H^2} \]

where:
- \(L\) is the length of the box,
- \(B\) is the breadth of the box,
- \(H\) is the height of the box.

### Step-by-Step Solution:

1. **Identify the dimensions of the box**:
- Length \(L = 20 \, \text{cm}\)
- Breadth \(B = 20 \, \text{cm}\)
- Height \(H = 10 \, \text{cm}\)

2. **Substitute the dimensions into the diagonal formula**:
\[ d = \sqrt{L^2 + B^2 + H^2} = \sqrt{20^2 + 20^2 + 10^2} \]

3. **Calculate the squares of the dimensions**:
- \(20^2 = 400\)
- \(20^2 = 400\)
- \(10^2 = 100\)

4. **Add the squares together**:
\[ d = \sqrt{400 + 400 + 100} = \sqrt{900} \]

5. **Calculate the square root**:
\[ d = 30 \, \text{cm} \]

Thus, the length of the longest rod that can be placed in the box is **30 cm**.